Automorphism group functors of algebraic superschemes (Q6585674)

Let \(k\) be a base field of zero or odd characteristic. By a superalgebra the author means a \(\mathbb{Z}/2\mathbb{Z}\)-graded supercommutative algebra over \(k\). An affine superscheme is a functor from the category of superalgebras to the category of sets, that can be represented by a superalgebra. Let \(X\) be a superscheme (so \(X\) has an open covering by affine supersubschemes). Denote by \(\mathbb{X}\) the \(k\)-functor of points of \(X\). The automorphism group functor \(\mathfrak{Aut}(\mathbb{X})\) is defined as follows: for a superalgebra \(A\) we have \(\mathfrak{Aut}(\mathbb{X})(A)=\{\text{invertible natural transformations of the functor }\mathbb{X}_A\}\). The main result of the paper is that if \(X\) is a proper superscheme, then \(\mathfrak{Aut}(\mathbb{X})\) is a locally algebraic group superscheme. This generalizes the theorem of Matsumura and Oort asserting that the automorphism group functor of a proper scheme is a locally algebraic group scheme. Moreover, it is also proved that if \(X\) is a proper superscheme with \(H^1(X,\mathcal{T}_X)=0\), where \(\mathcal{T}_X\) is the tangent sheaf of \(X\), then the group superscheme \(\mathfrak{Aut}(\mathbb{X})\) is smooth.